QI) Find equations of the Normal plane and Osculating plane of the curve x = 2 sin 3t, y = 2t, z = -2cos 3t at the point (0, 7,-2).

Q: Find the equation for (a) the tangent plane and (b) the normal line at the point Po(2,0,2) on the…

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Q: Find equations for the horizontal and vertical A)y= at (1, 똥) B) z = 0 at (0, ) D) y = 2 at (2, *)…

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Q: Find the parametric equations for, the tangent line to the curve It)=4?+1) i t (t+ 4)Ĵ ť 3+3 R at…

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Q: A variable straight line is drawn through the point of intersection of the straight lines x/a+y/b=1…

A: Given lines arexa+yb=1               ....(1)and  xb+ya=1               ....(2)Solving (i)and(ii) we…

Q: A variable straight line is drawn through the point of intersectio straight lines x / a+y/b = 1 and…

A: Given lines arexa+yb=1               ....(1)and  xb+ya=1               ....(2)Solving (i)and(ii) we…

Q: Find parametric equations for the normal line to the surfacez=ln(3x2 +7y2 +1)  at the point…

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Q: Consider the line in R? given by the parametric equation (a, y) = (2, -3) + t(-2, -9) What is the…

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Q: Find the point(s) of intersection of the curves r = 10 and r = 20 cos e. O A. (10,5) 3 OB.…

A: Given: r = 4  r = 20 cos θ

Q: Find an equation of the tangent(s) to the curve with parametric equations x = 1+ vt, y = e at the…

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Q: 7) A parametric curve is defined by the equations r = 2t + 3 and y = }t – 2 for -1<t < 3. (a) Make a…

A: Consider the values, t=-1, 0, 1, 2, and 3 for constructing the table.

Q: a) Find the equation of tangent line of curve that is given by parametric equations x = 3cost + sint…

A: The slope of the tangent at a point on the curve, y=f(x) is equal to the first derivative of the…

Q: Let I be the line with parametric equations x = -3-t y = 3 z = −3+t Find the shortest distance d…

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Family of Curves

A family of curves is a group of curves that are each described by a parametrization in which one or more variables are parameters. In general, the parameters have more complexity on the assembly of the curve than an ordinary linear transformation. These families appear commonly in the solution of differential equations. When a constant of integration is added, it is normally modified algebraically until it no longer replicates a plain linear transformation. The order of a differential equation depends on how many uncertain variables appear in the corresponding curve. The order of the differential equation acquired is two if two unknown variables exist in an equation belonging to this family.

XZ Plane

In order to understand XZ plane, it's helpful to understand two-dimensional and three-dimensional spaces. To plot a point on a plane, two numbers are needed, and these two numbers in the plane can be represented as an ordered pair (a,b) where a and b are real numbers and a is the horizontal coordinate and b is the vertical coordinate. This type of plane is called two-dimensional and it contains two perpendicular axes, the horizontal axis, and the vertical axis.  

Euclidean Geometry

Geometry is the branch of mathematics that deals with flat surfaces like lines, angles, points, two-dimensional figures, etc. In Euclidean geometry, one studies the geometrical shapes that rely on different theorems and axioms. This (pure mathematics) geometry was introduced by the Greek mathematician Euclid, and that is why it is called Euclidean geometry. Euclid explained this in his book named 'elements'. Euclid's method in Euclidean geometry involves handling a small group of innately captivate axioms and incorporating many of these other propositions. The elements written by Euclid are the fundamentals for the study of geometry from a modern mathematical perspective. Elements comprise Euclidean theories, postulates, axioms, construction, and mathematical proofs of propositions.

Lines and Angles

In a two-dimensional plane, a line is simply a figure that joins two points. Usually, lines are used for presenting objects that are straight in shape and have minimal depth or width.